(Indeterminate, like me. Think outside the box, but when you step outside the box ... try to keep one foot in)

Monday, April 11, 2011

The Equation That Couldn't Be Solved

What do Bach's compositions, Rubik's Cube, the way we choose our mates, and the physics of subatomic particles have in common? All are grounded by laws of symmetry, which elegantly unify scientific and artistic principles. Yet the mathematical language of symmetry - known as group theory - did not emerge from the study of symmetry at all, but from an equation that couldn't be solved.

For thousands of years mathematicians solved progressively more difficult algebraic equations, until they encountered the quintic equation, which resisted solution for three centuries. Working independently, two great prodigies ultimately proved that the quintic cannot be solved by a simple formula. Those geniuses, a Norwegian named Niels Henrik Abel and a romantic Frenchman named Evariste Galois, both died tragically young. Their incredible labor, however, produced the origins of group theory.

The first extensive, popular account of the mathematics of symmetry and order, The Equation That Couldn't Be Solved, by Mario Livio, is told now through abstract formulas but in a beautifully written and dramatic account of the lives and the work of some of the greatest and most intriguing mathematicians in history.

Contents

Finding the roots of a polynomial—values of x which satisfy such an equation—in the rational case given its coefficients has been a prominent mathematical problem.

Solving linear, quadratic, cubic and quartic equations by factorization into radicals is fairly straightforward, no matter whether the roots are rational or irrational, real or complex; there are also formulae that yield the required solutions. However, there is no formula for general quintic equations over the rationals in terms of radicals; this is known as the Abel–Ruffini theorem, first published in 1824, which was one of the first applications of group theory in algebra. This result also holds for equations of higher degrees. An example quintic whose roots cannot be expressed by radicals is x5 − x + 1 = 0. This quintic is in Bring–Jerrard normal form.

As a practical matter, exact analytic solutions for polynomial equations are often unnecessary, and so numerical methods such as Laguerre's method or the Jenkins-Traub method are probably the best way of obtaining solutions to general quintics and higher degree polynomial equations that arise in practice. However, analytic solutions are sometimes useful for certain applications, and many mathematicians have tried to develop them.

]Solvable quintics

Some fifth-degree equations can be solved by factorizing into radicals, for example , which can be written as or which has as solution. Other quintics like cannot be solved by radicals. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise togroup theory and Galois theory. Applying these techniques, Arthur Cayley has found a general criterion for determining whether any given quintic is solvable.[1] This criterion is the following.[2]

Both equations are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial P2 − 210zΔ, named Cayley resolvent, has a rational root in z, where

is solvable by radicals if and only if either a = 0 or it is of the following form:

where μ and ν are rational. In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,

The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression

where

and using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second. It is then a necessary (but not sufficient) condition that the irreducible solvable quintic

with rational coefficients must satisfy the simple quadratic curve

for some rational a,y.

The substitution c = − m / l5, e = 1 / l in Spearman-Williams parameterization allows to not exclude the special case a = 0, giving the following result:

If a and b are rational numbers, the equation x5 + ax + b = 0 is solvable by radicals if either its left hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers l and m such that

.

[Examples of solvable quintics

A quintic is solvable using radicals if the Galois group of the quintic (which is a subgroup of the symmetric group S(5) of permutations of a five element set) is a solvable group. In this case the form of the solutions depends on the structure of this Galois group.

A simple example is given by the equation , whose Galois group is the group F(5) generated by the permutations "(1 2 3 4 5)" and "(1 2 4 3)"; the only real solution is

However, for other solvable Galois groups, the form of the roots can be much more complex. For example, the equation has Galois group D(5) generated by "(1 2 3 4 5)" and "(1 4)(2 3)" and the solution requires more symbols to write. Define,

In general, if an equation P(x) = 0 of prime degree p with rational coefficients is solvable in radicals, there is an auxiliary equation Q(y) = 0 of degree (p-1) also with rational coefficients that can be used to solve the former.

However, it is possible some of the roots of Q(y) = 0 are rational (like in the F(5) example given above) or some as zero, like the solvable DeMoivre quintic,

where the auxiliary equation has two zero roots and is essentially just the quadratic,

such that the five roots of the DeMoivre quintic are given by,

where ωk is any of the five 5th roots of unity. This can be easily generalized to construct a solvable septic and other odd degrees, not necessarily prime.

Here is a list of known solvable quintics:

There are infinitely many solvable quintics in Bring-Jerrard form which have been parameterized in preceding section.

Up to the scaling of the variable, there is exactly five solvable quintics of the shape x5 + ax2 + b, which are[4] (where s is a scaling factor):

x5 − 100s3x2 − 1000s5

x5 − 5s3x2 − 3s5

x5 − 5s3x2 + 15s5

x5 − 25s3x2 − 300s5

Paxton Young (1888) gave a number of examples, some of them being reducible, having a rational root:

x5 − 10x3 − 20x2 − 1505x − 7412

x5 + 20x3 + 20x2 + 30x + 10

Solution:

x5 + 320x2 − 1000x + 4288

Reducible: −8 is a root

x5 + 40x2 − 69x + 108

Reducible: −4 is a root

x5 − 20x3 + 250x − 400

x5 + 110(5x3 + 60x2 + 800x + 8320)

x5 − 20x3 − 80x2 − 150x − 656

x5 − 40x3 + 160x2 + 1000x − 5888

x5 − 50x3 − 600x2 − 2000x − 11200

x5 + 110(5x3 + 20x2 − 360x + 800)

x5 − 20x3 + 320x2 + 540x + 6368

Reducible : -8 is a root

x5 − 20x3 − 160x2 − 420x − 8928

Reducible : 8 is a root

x5 − 20x3 + 170x + 208

An infinite sequence of solvable quintics may be constructed, whose roots are sums of n-th roots of unity, with n = 10 k + 1 being a prime number:

x5 + x4 − 4x3 − 3x2 + 3x + 1

Roots:

x5 + x4 − 12x3 − 21x2 + x + 5

Root:

y5 + y4 − 16y3 + 5y2 + 21y − 9

Root:

y5 + y4 − 24y3 − 17y2 + 41y − 13

Root:

y5 + y4 − 28y3 + 37y2 + 25y + 1

Root:

There are also two parameterized families of solvable quintics: The Kondo–Brumer quintic,

2 comments:

Yeah, interesting that the further members of a series of given kind are different in principle from the lower ones. It goes to show, we have right to logically assume continuation of previous attributes.

About Me

My weblog is named "Multiplication by Infinity", because "Division by Zero" was taken ... and "Division by Infinity" makes me feel very small ... Steven Colyer's Musings in Mathematical Physics and its Effects on Humanity and other Lifeforms.... And Pure Mathematics, Computer Science, Applied Mathematics, Experimental Physics, Engineering, Astronomy (not Cosmology so much), Space Exploration and Lunar Colonization.
I am a Rutgers 1979 Mechanical Engineer (Pi Tau Sigma) and Rutgers 1989 MBA.
("I study Politics and War that my children may study Mathematics and Philosophy."
- 2nd U.S. President John Adams)
I've already studied enough Politics and War and Economics for one lifetime, and so it's time for Math and Science